John Stuart Mill claimed that there are no a priori truths at all.
Rather, he argued, all truths (even truths of reason) are empirical truths
and as such all of our knowledge is based in experience.

Let's call this view empiricism about the truths of reason (or
of the a priori).

Let's first step back and look at the two different camps of epistemology
on this issue.

1. First, we have the empiricists. The most
famous empiricists in the history of philosophy are John Locke, George Berkeley,
and David Hume (athough the nature of Berkeley's empiricism is somewhat
complicated). According to the empiricists, experience is the basis of all
of our knowledge except perhaps analytic truths, which are seen as
"logical truths".

Some empiricists deny the a priori in only some respects. We can
call such empiricists moderate empiricists. According to such an empiricists,
analytic a priori truths are acceptable, but synthetic a
priori truths are not. The synthetic truths are really just
empirical truths.

A more radical empiricist such as Mill would contend that there
are no a priori truths at all -- even what we have been
calling analytic truths are really just empirical truths (hence
all knowledge is based on experience).

2. Second, we have the rationalists. The most
famous rationalists in the history of philosophy are Gottfried Leibniz,
Spinoza, and Rene Descartes. According to the rationalists, reason alone
is the foundation for all knowledge. According to some more radical rationalists
-- such as Leibniz -- all truths are analytic a priori and as such
can be known through conceptual containment (even a statement such as "Dr.
Panza's car is silver" can be shown to be true through conceptual containment,
and as such is no different than a statement such as "all bachelors are
unmarried"). So for a radical rationalist, there are no empirical
truths.

II. Empiricism and the Genesis and Confirmation
of Arithmetic Beliefs

It seems true that empiricism regarding the a priori seems more
reasonable with respect to synthetic a priori propositions (like
"nothing can be both red and green at the same time") than the analytic
a priori (this is because analytic statements simply seem to
refer to conceptual relationships, which don't appear to require checking
experience in any way at all). The synthetic statements, however,
do seem to be referring to the world of experience in a way that the
analytic statements do not. Let's take a look at some ways in which
empiricists might argue that synthetic statements are actually empirical.

The first example starts with the claim that arithmetic propositions are
synthetic a priori. This view is derived from Kant, who
uses this famous example in the Critique of Pure Reason, arguing
that the proposition "7 + 5 = 12" is synthetic a priori.

How does one know that this proposition is true?

Let's compare it with our other candidate synthetic a priori proposition,
"Nothing can be both red and green at the same time".

What is different about them?

Recall that the truth of the proposition "nothing can be red and green
at the same time" seemed to rely upon what we called conceptual exclusion
(claiming that something could not be both red and green was analyzed
as claiming that the concept of red excludes other color shades). But the
mathematical proposition doesn't seem to work like that. We don't say "7
+ 5 = 11" is false because the concept of "7 + 5" excludes "11" (at least
not in the way we meant).

So there seems to some real differences here. As such, some have argued that
7+5 = 12 (or any other arithmetic proposition) cannot be synthetic.

If so, how do we know that 7 + 5 = 12?

Some have argued that arithmetic develops from a base of simple observations.
I notice -- by using and manipulating objects in the world -- that everytime
I put 5 objects with 7 objects, I wind up with 12 objects.
If this is how we know the proposition 7 + 5 = 12 - and it
does seem plausible that such empirical work may be at the basis of learning
arithmetic, then there will be a large problem that needs to be reconciled.

The problem is this: it is argued that since 7 + 5 = 12 is synthetic a
priori, it is necessarily true.

If experiences with objects is really the basis for our knowledge of mathematical
propositions, then no mathmatical proposition learned in such a way will
be necessarily true. The reason is this: it surely seems conceivable that
there is some possible world where -- because of some freak law of nature
-- everytime 5 objects are added to 7 objects, the new set is always 11 objects
big. Since mathematic propositions are justified by empirical observation
with objects, the claim that "7 + 5 = 12" would not be true in that possible
world. If it is not true in some possible world, then it is not necessarily
true.

However, there is a counter-reply from the classical view. It asks:

Is the genesis of a belief always what justifies that belief?

In other words, even if physical counting is how we come to have mathematical
beliefs (it serves as their genesis), does that mean that physical counting
should be the justification for those beliefs? Perhaps this is not
the case. Perhaps we would say that physically counting 7 objects and then
5 objects added to it is not a test of the proposition 7 + 5 = 12,
but rather an exemplification of that proposition.

What does this mean? Let's think about what it means to exemplify.

Let's say you tell me "touching the stove when hot causes pain". And then
I touch it when hot, and I feel a great amount of pain. The pain I feel
gives evidence for what the proposition is saying. At the same time,
however, it serves as proof of the proposition when it does give evidence
for it. If every time someone touched a hot stove nothing happened, this
would be evidence that the proposition is false. This is how contingent
empirical propositions work. Empirical propositions are about the
world. So when the world fails to serve as evidence for a proposition,
the proposition has been falsified and no longer has any reason to be believed.

Next, let's say that I say "7 + 5 = 12". And then I show you what this means
by placing 5 oranges with 7 oranges, and counting them to total 12 oranges.
The difference here is that showing you the oranges isn't evidence
in the way that touching the stove is; rather, this show merely exemplifies
what the proposition means. Truths of reason are not propositions
about the world and so the world does not serve as evidence
for them, although it can serve as an exemplification for what
the proposition means.

So if I added 7 oranges to 5 oranges and got 11 oranges, I would not say
"7 + 5 isn't 12" but rather "ah, here it seems that the world does not exemplify
the proposition".

If arithmetic propositions are not about the world, what are they
about?

The classical theorist will say that the way the world exemplifies our
beliefs may in fact change, while the status of the beliefs (which are no
longer exemplified) stays true. Clearly, a classical theorist will say, it
is not about the objects we are adding. Rather, it is about the nature of
certain mathematical concepts (numbers). The fact that objects do not
exemplify what we know to be true about numbers doesn't
falsify the proposition about numbers. It is the case that -- conceptually
speaking -- adding the number 7 to the number 5 yields the number 12, even
if it is not the case that adding 7 oranges to 5 oranges
equals 12 oranges (because this is a proposition concerning something
else, namely the nature of oranges in the world).

Thus the classical view could argue that the proposition "7 oranges + 5
oranges = 12 oranges" is contingent and empirical (since it is about
the world, which could be different) whereas the proposition "7 + 5 = 12"
is necessary and synthetic a priori (since it is about concepts,
which cannot be different, and the truth of the proposition is not known
via conceptual containment).

To counter, however, the radical empiricist will simply claim that there
are no abstract entities such as numbers. If there are no "abstract
numbers" then all that is left for an arithmetic proposition to be about
is the world. All that really exist are things in the experienced world
(abstract concepts such as numbers are not "experienced"). So propositions
that look like they are about numbers are really just generalizations
about the behavior of physical objects. In other words, upon counting many
collections of 7 objects and 5 objects, we always end up with 12 objects,
and then afterwards we generalize to say that 7 and 5 is 12 (and leave
out the talk about objects).

Thus for a radical empiricist arithmetic propositions are contingent and
empirical.

III. Empiricism and Logical and Analytic Truths

In the former section we looked at possible empiricist
attacks on existence of synthetic a priori truths (such as those
purported to be in mathematics). But a radical empiricist will deny
more -- they will claim that even the analytic a priori truths are
empirical (and thus contingent).

Let's take the example of "all vixens are female".

Let's say that after a number of years, scientists start to recognize that
vixens actually have a number of biochemical features that are typically
associated with being male. In fact, after a number of more years so many
male-confirming features arise that there begins to be doubt about whether
vixens really are female at all. Doesn't this show that the proposition "all
vixens are female" can be falsified, and so thus is an empirical proposition?

The classical response to this will be to say that in such a situation,
we have really just created a new word with a new meaning as opposed to keeping
the old word but changing it's meaning. This sounds odd, but let's say:

Pre-2003 the word "vixen" is used to mean "female fox".

However, after 2003 experiments show that they are really males. So

Post 2003 the word "vixen" is used to mean "male fox".

The claim here will be that it is still analytic that "the pre-2003
word vixen" means "female fox". Notice that we have two words here
-- "vixen" as used before 2003, and "vixen" as used after 2003. Post 2003
we use a sound that is identical to the one we used before 2003, but it
really is a new word with a new meaning.

As such, the classical view will say that analytic truths cannot be falsified
at all, and so are not empirical.